Question

Obtain the partial differential equation by eliminating the arbitrary constant from the relation

x^2/a^2 +y^2/b^2 +u^2/c^2 =1

Answer 1

$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{u^2}{c^2} = 1$

Differentiating w.r.t. x

$\frac{2x}{a^2} + \frac{2u}{c^2}\frac{du}{dx} = 0 \;\;\;(1)$

Again differentiating (1) w.r.t. x

$\frac{2}{a^2} + \frac{2ux}{c^2} \frac{d^2u}{dx^2} + \frac{2}{c^2} (\frac{du}{dx})^2 = 0$

Multiplying by x

$\frac{2x}{c} + \frac{2ux}{c^2} \frac{d^2u}{dx^2} + \frac{2}{c^2} (\frac{du}{dx})^2x = 0$

From (1):

$\frac{2x}{a^2} = \frac{-2u}{c^2} \frac{du}{dx} \\ \frac{-2u}{c^2} \frac{du}{dx} + \frac{2ux}{c^2} \frac{d^2u}{dx^2} + \frac{2x}{c^2} (\frac{du}{dx})^2 = 0$

Put

$\frac{du}{dx} = p \\ \frac{d^2u}{dx^2} = s \\ -up + uxs + xp^2 =0$